Many cancer studies (animal, clinical and epidemiological) lead to event- time data in which multiple events, possibly of different types can occur to each subject or to each of a group of related subjects, for example animals born in the same litter. Examples of such data include time to tumor detection, time from remission to relapse into an acute disease phase, and times to discontinuation of experimental medications. Methods for the statistical analysis of such data will need to take into account heterogeneity between subjects or between groups of related subjects. This can be achieved by the incorporation of additional random effects into the standard survival models. The present research focusses on models involving "frailties", unobserved random proportionality factors applied to the hazard function. Methods of parametric, semiparametric and nonparametric estimation in such models will be investigated and applied to multiple event data arising from studies of cancer and other diseases. Bayesian methods, often involving the use of the Gibbs sampler will be considered as well as classical frequentist techniques. Emphasis will be placed on the formulation of models in such a way that covariate effects can be interpreted both conditionally on the random effect, and unconditionally. Frailty models will be used to investigate selection biases in epidemiologic studies and in the analysis of compliance data in clinical trials. Methods will be proposed for analysis of data when there is more than one plausible time scale, for example in a mortality study of asbestos workers one may want to use both calendar time and cumulative exposure to asbestos dust. In a study of the development of tumors in experimental animals, both the time from the start of the study and the time since the last tumor may be relevant. A general class of multiplicative intensity models involving multiple time scales will be studied. Characterizations of multivariate' survival distributions will be explored with a view to deriving graphical methods of assessing goodness of fit. The performance of a new algorithm for estimating a bivariate survival distribution subject to a general pattern of censoring will be investigated.